P(0 heads) = (1/2)^16
P(1 head) = (# of ways to choose 1 coin from 16) * (1/2)^16
There's this thing called factorial. The factorial of a number is that number times one less than that number, times two less than that number, etc., all the way down to 1. The symbol for factorial is !.
Example: 16! = 16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1
There's this thing called combinations. If you want to pick 1 coin from 16 coins, the number of ways you can do that is the number of ways you can arrange all of the coins (16!) divided by the number of ways you can arrange the coin you do want (1!) times the number of ways you can arrange the other coins you don't want (15!). Combinations are written as C(x,y) or x nCr y. Combinations are said as 'x choose y.'
Example: 16 choose 1 = 16!/(15!*1!) = 16*15*14* . . . / (15*14*13* . . . *2*1)(1) = 16
For this problem we're flipping 16 coins, so there are 2^16 possible outcomes.
Probability = # ways what we want can happen / # of ways everything can happen.
P(0 coins are heads when tossing 16 coins) = (16 choose 0) / 2^16
P(0 heads) = ( 16!/(16!*0!) ) / 2^16 = 1 / 65536
P(1 head) = ( 16!/(15!*1!) ) / 2^16 = 16 / 65536
P(2 heads) = ( 16!/(14!*2!) ) / 2^16 = 120 / 65536
P(3 heads) = ( 16!/(13!*3!) ) / 2^16 = 560 / 65536
P(4 heads) = ( 16!/(12!*4!) ) / 2^16 = 1820 / 65536
P(5 heads) = ( 16!/(11!*5!) ) / 2^16 = 4368 / 65536
P(6 heads) = ( 16!/(10!*6!) ) / 2^16 = 8008 / 65536
P(7 heads) = ( 16!/(9!*7!) ) / 2^16 = 11440 / 65536
P(8 heads) = ( 16!/(8!*8!) ) / 2^16 = 12870 / 65536
P(9 heads) = ( 16!/(7!*9!) ) / 2^16 = 11440 / 65536
P(10 heads) = ( 16!/(6!*10!) ) / 2^16 = 8008 / 65536
P(11 heads) = ( 16!/(5!*11!) ) / 2^16 = 4368 / 65536
P(12 heads) = ( 16!/(4!*12!) ) / 2^16 = 1820 / 65536
P(13 heads) = ( 16!/(3!*13!) ) / 2^16 = 560 / 65536
P(14 heads) = ( 16!/(2!*14!) ) / 2^16 = 120 / 65536
P(15 heads) = ( 16!/(1!*15!) ) / 2^16 = 16 / 65536
P(16 heads) = ( 16!/(0!*16!) ) / 2^16 = 1 / 65536